Let $\sum _ { k \geqslant 0 } c _ { k } x ^ { k }$ be a power series with radius of convergence $R > 0$. We set
$$\forall x \in ] - 1,1 [ , \quad g ( x ) = \sum _ { k = 0 } ^ { + \infty } x ^ { k }.$$
Show that $g$ is of class $\mathcal { C } ^ { \infty }$ on $] - 1,1 [$ and that
$$\forall j \in \mathbb { N } , \quad \forall x \in ] - 1,1 [ , \quad g ^ { ( j ) } ( x ) = \frac { j ! } { ( 1 - x ) ^ { j + 1 } }.$$